Theorem pwpwab 3960

Description: The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
pwpwab	|- ~P ~P A = { x | U. x C_ A }

Distinct variable group:   x, A

Proof of Theorem pwpwab

Step	Hyp	Ref	Expression
1		vex 2733	. . 3 |- x e. _V
2		elpwpw 3959	. . 3 |- ( x e. ~P ~P A <-> ( x e. _V /\ U. x C_ A ) )
3	1, 2	mpbiran 935	. 2 |- ( x e. ~P ~P A <-> U. x C_ A )
4	3	abbi2i 2285	1 |- ~P ~P A = { x | U. x C_ A }

Syntax hints:   = wceq 1348   e. wcel 2141   {cab 2156   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566   U.cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-uni 3797

This theorem is referenced by:  pwpwssunieq  3961